Question

The figure shows two 4-tile patterns.
Either one or both of the patterns can be used any number of times and in any
orientation to construct a new pattern. Which one of the options below cannot be
constructed by using only these two 4-tile patterns assuming there are no overlaps
among them?

• A. Option A shown in the figure
• B. Option B shown in the figure
• C. Option C shown in the figure
• D. Option D shown in the figure

Hint:

Try mentally tiling each option using rotations and mirror images of the two 4-tile patterns; the option that always leaves an unfillable gap is the one that cannot be constructed.

Answer 1

The Correct Option is C

Step 1: The two given 4-tile patterns are made of unit squares joined edge-to-edge. Any new figure must be built entirely from copies of these two patterns, rotated or reflected as needed, placed with no overlaps and no gaps.
Step 2: Since each base pattern covers exactly 4 unit tiles, any constructible figure must have a total tile count that is a multiple of 4 - a necessary first check.
Step 3: For shapes that pass the area check, examine whether the figure's boundary and internal corners can be partitioned into whole copies of the two patterns in some rotation/reflection. A shape fails if it creates a notch, a single isolated tile, or a corner profile that neither tetromino-like pattern (in any of its 8 orientations) can fill.
Step 4: Testing all four options this way, Option A, Option B, and Option D can each be cleanly decomposed into whole copies of the two patterns.
Step 5: Option C, however, contains a region whose boundary shape cannot be matched by either pattern in any rotation or reflection - no valid tiling of it exists using only the two given patterns.
Final Answer: \(\boxed{\text{Option C}}\)